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NASA
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CONTRACTOR v.7
REPORT
FORE-AND-AFT STIFFNESS CHARACTERISTICS OF PNEUMATIC TIRES
by R. N. Dodge, David Ome, mzd ,$. K. Chrk
Prepared by
THE UNIVERSITY OF MICHIGAN
Ann Arbor, Mich.
f Or
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, D. C. l OCTOBER 1967
https://ntrs.nasa.gov/search.jsp?R=19670029205 2018-06-09T15:43:31+00:00Z
TECH LIBRARY KAFB. NM
NASA CR-900
FORE-AND-AFT STIFFNESS CHARACTERISTICS
OF PHEUMATIC TIRES
By R. N. Dodge, David Orne, and S. K. Clark
Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.
Issued by Originator as Technical Report No. 6
Prepared under Grant No. NsG-344 by THE UNIVERSITY OF MICHIGAN
Ann Arbor, Mich.
for
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151 - CFSTI price $3.00
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS
I. INTRODUCTION
II. SUMMA.RY
III. ANALYSIS
IV. COMPA.RISON OF THEORY WITH EXPERIMENT
v, APPENDIX I
VI. APPENDIX II
VII. A.CKNOWL,EDGMFdTS
vm. REFERENCES
iii
Page
v
1
3
4
:11
21
31
33
34
'LIST OF ILLUSTRATIONS
Table Page
I.
II.
III.
Figure
1.
20
3.
4.
50
6”
7.
8.
9.
10.
11.
12.
13.
14.
15.
Summary of Geometric, Elastic, and Structural Properties of Five Automotive Tires
Summary of Experimental and Calculated Values of Kf
Comparison of Results Using Approximate Moduli
Idealized model of pneumatic tire for analyzing fore-and-aft spring rates.
Loaded element of portion of the model
Meridional cross-section of the tire showing symbols used.
Experimental apparatus for checking KS.
Load-.deflection data for double tube experiment used to confirm the expression for KS,
Idealized mid-line profiles of five pneumatic tires.
Photograph of experimenf,a.l apparatus for KfO
Schema.t,ic of fore-and-a.ft. spring-ra.te test.
Fore-and-aft load-8deflection curves. Tire No. 1.
Fore-and-aft load-deflection curves. Tire No. 2.
Fore-and-aft load-deflection curves. Tire No. 3.
Fore-and-aft load-deflection curves. Tire No. 4.
Fore-and-aft load-deflection curves. Tire No. 5.
Comparison of exact and approximate circumferential modulus Ego
Comparison of exact and approximate shear modulus G. .
V
13
13
26
10
12
14
14
16
17
18
19
20
22
24
LIST OF ILLUSTRATIONS (Continued)
Figure Page
16. Sidewall location for effective G. 26
17. Description of hypothetical tire for example problem. 28
vi
I. INTRODUCTION
Pneumatic tires are functional parts of many dynamic systems. In
order to effectively design and engineer such systems, it is often neces-
sary to know the mechanical properties of the component parts, including the
tires. One of the major roles of this research group has been to study and
analyze some of the important mechanical properties of pneumatic tires and
to present rational methods for predicting them.
Several paths have been followed by thi s research group in developing
approximate methods for predicting mechanical properties of tires. One
technique involved modeling of the penumatic tire as a cylindrical shell
supported by an elastic foundation. 5-7 This model gives relations for pre-
dicting several properties involving deformation in the plane of the wheel,
such as contact patch length VS. vertical deflection, vertical load vs.
vertical deflection , plane vibration characteristics, transmissability
characteristics, and dynamic response to a point load.
4 A more recent attempt involved analyzing the pneumatic tire as a
string on an elastic foundation. This model is primarily used to predict
lateral stiffness characteristics, vertical stiffness characteristics and
twisting moments. However, it has been found to be useful only when the
inflation pressure is high, such as in the case of aircraft tires.
This report presents a method for predicting the fore-and-aft stiffness
characteristics of pneumatic tires. Fore-and-aft properties are important
in the overall analysis of a tire since they represent the contributions of
1
the carcass and tread to braking and tractive elasticity. A different mcdel
is required here since neither the cylindrical shell nor the string on the
elastic foundation provide means for transmitting such loads. It is hoped
that this model will prove satisfactory for predicting fore-and-aft charac-
teristics of various tire designs.
2
II. SUMMARY
An elastic bar supported by a foundation exhibiting elasticity in shear
serves as a model for determining the fore-and-aft stiffness properties of a
pne,umatic tire. The differential equation representing the deformation is
derived and solved, and the resulting solution gives a means for calculating
a fore-and-aft spring rate for the model.
A series of five tires of various sizes and structures was used for test-
ing the validity of the proposed model. A set of static tests was performed
to establish an experimental value for the fore-and-aft spring-rate for the
various tires. Addit,ional structural data, required by the ana1ytica.L solu-
tion of the model, were also obtained from the tires. A comparison of the
calculated and experimental results was reasonably satisfactory, indicating
that the proposed model can be used to roughly approximate fore-and-aft stiff-
ness characteristics.
A complete tabu.Lar summary of the geometry and composition of the five
tires is included for easy reference. All experimentma and analyticaIL results
are summarized and compared in graphical form.
3
III. ANALYSIS
To represent fore-and-aft stiffness characteristics, the pneumatic tire
is idealized as an elastic bar supported by an elastic shear foundation (see
Figure 1). The elastic bar portion of the model represents the tread region
of the tire which is loaded by the fore-and-aft load F. In addition to the
restraint offered by the stiffness of the tread region itself, resistance to
deformation by the load F is provided by the tires' ability to withstand
shearing forces in the sidewall regions. This portion of the tire is repre-
sented in the model by the elastic shear foundation.
. . RIM
Figure 1. Idealized model of pneumatic tire for analyzing fore-and-aft spring rates.
If it is assumed that the restraining force of the elastic shear founda-
tion is directly proportional to the displacement, an elemental segment of
the elastic bar can be set in equilibrium as shown in Figure 2. Note that ad-
4
vantage is taken of the symmetry present in the model.
F/2 --
( Ksw)dx
--j kS+gdx
dx
Figure 2. Loaded element of a portion of the model.
In Figure 2, S is the force acting on the bar, u is the displacement,
and KS is the spring rate per unit length of the shear foundation. It is
assumed that KS is provided only by the shear resistance of the sidewall.
From Figure 2, it is seen that one may approximate
where G is the effective shear modulus of the sidewall, H is the sidewall
thickness, and A is the length along the sidewall from the rim to the point
of intersection of the tread and carcass.
From equilibrium of the element,
S = TA, = EeAs = A,E & ax
5
where T is the stress, e the resulting strain, A, the cross-sectional area of
the bar at any location, and E the effective extension modulus of the tread
region in the circumferential direction. Thus,
2 - - q2u au = 0
2 ax (1)
where
K. q2 = -&
S
The general solution of this equation is
u = Cl cash qx + C2 sinh qx (2)
The boundary conditions for this problem are determined by assuming that
each half of the tire (fore-and-aft of the contact patch) is equally loaded,
so that
at x = 0, s=$
at x = Jta, s = 0
Substituting (3) into (2) gives
Cl = -F . 2A,E tanh sraq '
c,=-2- 2&N
Thus,
u(x) = & C sinh qx - cash qx S 1 tanh qJta
(3)
(4)
(5)
6
The fore-and-aft stiffness is determined by finding the ratio of the applied
load tc the displacement at the point of application of the load. Thus,
Kf = 1 --$J 1 = I- 2A,Eq (tanh qKa> 1 (6)
Equation (6) now represents a relationship for the fore-and-aft spring-rate
of a pneumatic tire idealized as an elastic bar supported by a shear founda-
tion. As can be seen From Eqs. (1) and (6), the application of Eq. (6) to a
real tire requires a knowledge of the effective stiffness (AsE) of the tread
region in the circumferential direction., the effective shear modu:Lus of the
side -wall region G,the effect,ive sidewall t.hickness H, and the length along
the mean meridional section from t.he rim to the intersection of the tread and
carcass, A.
The extension modulus in the circumferential direction and the shear
moduPus of the carcass usually vary from one location to anot,her in the merid-
iona. direction because of t.he orthotropic nature of the tire carcass, SC some
criteria must be established to compute A,E and G for a given tire section,
Both the extension and shear modulus used for the resu.Lts Listed in this
.report were obtained by averaging the actual va.Lues of these properties
throughout the cross-section, This technique has been successful because
i,he variation in t,hese properties has not been too nonlinear. However, it
has a great disadvantage in a simplified analysis such as this because it
requires lengthy calculations which cannot be done efficiently without the
aid of a digital computer. Por this reason an effort. has been made to ob-
tain some simplified, but reasonably accurate., approximations for the A,E
7
and G necessary for calculating the fore-and-aft spring constant. The re-
sults of this effort are included in Appendix I and give a satisfactory ap-
proximate technique for calculating these properties.
Y
Shear Modulus G
Extension Modulu Normal to Plane of Paper Ee
Figure 3. Meridional cross-section of the tire showing symbols used.
In order to establish some validity for the assumption that the shear
foundation modulus KS can be estimated by considering shear effects only, a
simple experiment was performed by gluing a metal strip along the line of con-
tact of two rubber cylinders placed side by side (see Figure 4). A load was
attached to the bar and the resulting deflection was measured by the dial in-
dicator. The slope of the experimental load-deflection curve, related to Ks,
was then compared with the value of KS obtained from the relation given above,
A summary of this experiment is presented below:
8
Indicator
Figure 4. Experimental apparatus for checking KS.
B = 10.0 in. $ = 1.252 in.
dl = 0.986 in. H = 0.133 in.
el = 0.10 in. A = 1.72 in.
e2 = 0.15 in. G = 20013 lb/in.2
From the test data (Figure 5), the slope of the load-deflection curve
yields a KS = 22.0 lb/in./in. The calculated value for the double tube is
9
= 20.6 lb/in./in.
(A factor of 4 appears in this computation because of the double tube arrange-
ment.) The close comparison between the experimental KS and the calculated
one, assuming that the foundation is flat rather than curved, indicates that
any curvature effects are minor.
I I I I I .Ol .02 .03 .04 .05
DEFLECTION - IN.
Figure 5. Load-deflection data for double tube experiment used to confirm the expression for KS.
10
IV. COMPARISON OF THEORY WITH EXPERIMENT
In order to investigate the validity of Eq. (6), a series of static
fore-and-aft stiffness tests were run on representative tires. Before
reporting these tests and their results, the five tires used are described
in detail. The idealized centerline profiles of the tires are shown in
Figure 6. Tire No. 1 is a domestic 4-ply, 8.00 x 14 bias-ply tire with
standard nylon cord. Tire No. 2 is a 2-ply, 7.50 x 14 bias-ply tire with
standard nylon cord. Tire D;o. 3 is an imported 4-ply, 5.90 x 15 bias-ply
tire with nylon cord. Tire No. 4 is an imported 7.50 x 14 radial-ply tire
with overheads reinforced with wire cord. Tire No. 5 is a European made
155 mm x 15 in. radial-ply tire with overheads reinforced with nylon cord.
Table I is a summary of the pertinent elastic and geometric parameters
required from the five tires. Using the results in this table, Figure 6,
Eq. (6) and the proper elastic properties, it is possible to calculate the
fore-and-aft stiffness of the five tires. Carrying out these computations
gives the calculated values presented in Table II.
To check the accuracy of the calculated values, the five tires were
tested in the apparatus illustrated in Figures 7 and 8. In this testing
procedure the tires were loaded vertically to a fixed deflection. Then a
varying fore-and-aft load was applied and the resulting deflection recorded.
The slope of these load-deflection curves represents the experimental fore-
and-aft spring-rates. These tests were run for different vertical deflections
and inflation pressures. The results of these tests are summarized in Fig-
11
4
3 IN.
2
I
6
5
4
IN. 3
2
I
0 I 2 3 IN.
IN. 2-
I-
I I 0 I
iN.2 3
Figure 6. Idealized mid-line profiles of five pneumatic tires.
TABLE I
SUMMARY OF GEOMETRIC, ELASTIC, AND STRUCTURAL PROPERTIES OF FIVE AiJ'I0MOTIVE TIRES
Tire 1 Tire 2 Bias-Ply Bias-Ply
Tire 3 Bias-Ply
Tire 4 Tire 5 Radial-Ply Radial-Plv Item (Ref. 5, Table I)
- ~~ __ ~- A0 - outside radius of tire L - half circumference ET - extension modulus, treed rubber A - length, mea" meridional section H - effective thickness for K, G - effective shear modulus for K, AS - effective spring rate, circumfere"tial KS - spring rate, shear foundation BETAC - cord half angle, crown ROC - radial location, crown ROB - radial location, rim R - idealized radius, sidewall YC - y-coordinate, center for R xc - x-coordinate, center for R Rl - idealized radius, crown region ERUB - extension modulus, carcass rubber GRUB - shear modulus, carcass rubber MUFWB - Poisson ratio, carcass rubber WORD - shear modulus, cord AEsLJEc - spring rate, cord DIAMC. - effective diameter, cord MUC - Poisson ratio, cord TPLY - effective ply thickness NCORD - cord count, crown ALPSTR - normal angle, Intersection ALPHR - [email protected] angle, rim Bw - tread width
8.00x14 7.50x14 ~-- 5.90x15 7.50x14 - 15% x15" 12.525 13.94 12.875 13.44 39.35 't3.79 40.45 42.22
670. 560. 481. 690. 4.6408 5.8404 4.9217 5.7238 0.164 0.110 0.160 0.250
28440. 47164. 43876. 779. 747. 760.
269. (Gxy) 11906.
2010. 1747. 2852. 18.98 0.6458 0.6283 0.6109 0.3142
12.120 13.38 12.355 12.92 7.078 7.20 7.515 7.06 3.22 3.03 2.84 3.13 1.935 2.490 2.092 2.454
-0.057 0.465 -0.146 0.405 3.40 2.62 2.76 3.54
438. 310. 370. 625. 146. 103. l23. 208.
0.500 0.500 0.500 0.500 705. 705. 705. 705. 200. 623. 317. 350.
0.025 0.040 0.026 0.025 0.700 0.700 0.700 0.700 0.041 0.055 0.040 0.040
26. 19. 24. 18. 0.7746 0.6085 0.6665 0.6427 2.2148 2.5331 2.3973 2.4714 4.36 4.40 3.20 4.60
12.25 38.48
490. 4.5529 0.280
144. (G,,) 23027.
23.50 0.2356
11.70 7.59 2.17 1.939 0.704 4.88
X0. 100.
0.500 705. x7.
0.023 0.700 0.040
20. 0.5775 2.6792 3.68
TABLE II
SUMMARY OF EXPERIMENTAL ANDCALCULATED VALUES OF Kf
Tire Kf - lb/i".
Experimental Calculated
1 2530 2503 2 2300 2286 z 2780 2944
1340 1469 5 1265 950
13
Figure 7. Photograph of experimental apparatus for Kf.
Figure 8. Schematic of fore-and-aft spring-rate test.
14
ures 9 through 13.
In general it can be seen from these curves that the fore-and-aft spring-
rate increases only slightly with increasing vertical load and with increasing
internal pressure. Since Eq. (6) does not. account for the slight increase
due to these factors, a comparison between the experimental and calculated re-
sults must be made in a somewhat arbitrary fashion. However, since the experi-.
mental values are nearly the same for all conditions examined, any results used
as a comparison with the calculated values will serve as a meaningful check,,
The comparisons shown in Table II are based on experimental values obtained
from vertical tire deflections of one inch and by use of manufacturers .rat.ed
inflation pressure., The experimental values were determined by measuring the
s:Lopes in the linear portions of the load-deflection curves. These compari-
sons indicate that the simple model formulat-ed above gives a method for ap-
proximating the fore-,and-,aft spring-rate of pneumatic tires using only the
geometric, elastic, and structural properties required by mcst tire designers,
At the end of the Appendix II an examp'le problem is worked out, illustrating
how an approximate fore-and-aft spring rate can be calculated if' the correct
input data is available.
15
300 15 psi
0 l/2 in. vertical deflection X 3/4 in. vertical deflection A I in. vertical deflection q I l/4 in. vertical deflection
25 psi
X
X 0
Y en
0 X
f 0
E 0
x ; O
n n
I ix -
! x
I I I I I I I I I II I I I I .I I I I
0 .Ol .02 .03 .04 .05 .06 .07 .08 0 .Ol 02 .03 .04 .05 .06 .07 .08
HORIZONTAL DEFLECTION-IN.
35 psi
0 A
0 *
0 x A 0
0 x A 0
0
X
z 0
[ O
I I I I I I I I
.Ol .02 .03 .04 .05 .06 .07 08
Figure 9. Fore-and-aft load-deflection curves. Tire No. 1.
- F
18 psi
I I I I I I I I
0 .Ol .02 .03 04 .05 .06 .07 .08
0 l/2 in. vertical deflection X 3/2 in. vertical deflection A I in. vertical deflection q I 114 in. vertical deflection
28 psi 38 psi
B
I I I I I I I I I I I I I I I I I
.Ol .02 .03 04 .05 .06 .07 .08 0 .O I .02 .03 .04 .05 .06 .07 .08
HORIZONTAL DEFLECTION
Figure 10. Fore-and-aft load-deflection curves. Tire No. 2.
9 psi
0
’ A q A
0 I /2 in. vertical deflection X 3/4 in. vertical deflection A I in. vertical deflection o I l/4 in. vertical deflection
I9 psi
0 B 0 ; x
A
x X 0
B x 0 O
0 xe A0 a x 0
x 0 O 0 0
::
nx
L9 X
I I I I I I I I I I I I I I I I I
.Ol .02 .03 .04 .05 .06 .07 .08 0 -01 .02 03 04 .05 -06 07 .08
HORIZONTAL DEFLECTION
29 psi
0 B
0 A x A
0 X
0
A x 0
O x O A
0 x 0
- 0 X
F O 0
0
I I I I I
.d5 I I I
.Ol 02 .03 .04 .06 .07 .08
Figure 11. Fore-and-aft load-deflection curves. Tire No. 3.
-
I8 psi
II a
0 l/2 in. vertical deflectiol, X 3/4 in. vertical deflection A I in. vertical deflection o I l/4 in. vertical deflection
28 psi
0 .Ol .02 .03 .04 .05 .06 .07 .08 0 .Ol .02 .03 .W .05 .06 .07 .08
HORIZONTAL DEFLECTION - IN.
38 psi
& “0 O q 0
0 0
0 O 0
DI .02 .03 04 .05 .06 .07 .08
Figure 12. Fore-and-aft load-deflection curves. Tire No. 4.
IO psi
200-
o l/2 in. vertical deflection x IV4 in. vertical deflection A I in. vertical deflection o I l/4 in. vertical deflection
20 psi 30 psi
0 .Ol .02 .03 .04 .05 .06 ,07 .08 0 .Ol .02 03 .04 .05 .06 .07 .08 0 .Ol .02 .03 .04 .05 .06 .07 .08
HORIZONTAL DEFLECTION -IN
Figure 13. Fore-and-aft load-deflection curves. Tire No. 5.
v. APPENDIX I
This Appendix presents the results of an effort to simplify the compu-
tations involved in obtaining the extension modulus of the carcass in the
circumferential direction, E 8' and the shear modulus of the carcass, G.
Reference 1 gives exact expressions for the moduli of laminated ortho-
tropic two-dimensional sheets, and these are good representations for the
elastic constants of a tire carcass. However the expressions derived in
Ref. 1 are quite lengthy to evaluate. They can be simplified considerably
by considering the structure to be made of inextensible cords, so that the
modulus of elasticity E X’
parallel to the cords in a single ply, becomes
indefinitely large. By simplifying the expressions of Ref. 1 in this way,
one gets
EG = 4Gxy sin*a. cos*a + Ey(cos2Q: - sin2a)*
4 (7) sin 0
where a is the local cord half-angle at any meridicnal locaticn, G is the XY
shear modulus of an individual ply and E is Y
the extension modulus of an in-
dividual ply in the direction normal to the cords. This is a very gcod ap-
proximation for EQ as long as the cord half-angle is net less than 30". This
is shown in Figure 14 where Eq. (7) is compared with the exact formulation of
Ref. 1. Thus, Eq. (7) is a good approximation for an ordinary bias-construc-
tion tire, where the angle is almost always greater than 30".
An approximation for the shear modulus can be obtained from a strength
of materials analysis. This is reproduced in Appendix II in detail, and gives
21
10 \
\
\
\
\
\
\
\
Theoretical ( Ref. 1) ---- Approximate (Eq. 7)
E, = 1.2683 x lo5
0
Figure 14.
E, = 1370.7 F,,= 2.2557 x 10 5
G,,= 272.92
Cord Half - Angle ct”
Comparison of exact and approximate circumferential modulus EQ.
22
G = Exsin2a cos2ct (8)
where Ex is the extension modulus in the direction of the cords o!f an
individual ply of the-tire carcass material. This is a very good.approxima-
tion for the shear modulus except at cord half-angles near 0 and 90" as
shown in Figure 15 where the exa,ct shear modulus expression from Ref. 1
is compared with Eq. (8). Again, this is valid approximation for almost
all tire constructions since cord angles usually lie between 30" and 60".
Reference 2 presents a concise method for calculating s, 5,
and G XY
as a function of the geometric and elastic properties of a single
ply of the carcass material. The expressions for these moduli are listed
below:
Ex = (AEsUBC)(NC~RD)/TPLY
5 = ERU+ + 2.,&j
G XY
= 705 K'AHh, + GRUB(l-A, hH)
where:
K(DIAMC) h=---Tlm-
4 = K(DIAMC)(NCORD)
The constant k is an area coefficient equal to G. The other
parameters are defined in Table I of this report. Unfortunately both the
expression for E 8
and G are still functions of the cord half-angle, o,
which varies from position to position around the cross-section. However,
23
1
1(
10 3-
Theoretical 1 Ref. 1)
-
E, = 1.268 x lo5 E, = 1371 F,,= 2.256 x lo5 G,, = 273
11 10 45 60 75 90 Cord Half-Angle - 01~
Figure 15. Comparison of exact and approximate shear modulus G.
24
it has been found possible to select average Locations at which r.he values
of the extension modulus and shear modulus represent usefu.1 va:Lues for the
entire cross-sec.t.ion.
The extensiona. stiffness of the "bar" is represented by A,E and
primarily depends on the extensicna.1 stiffness of the carcass in the tread
shoulder region of t,he tire. Ihe cord half-angle in the tread and shculder
region of the tire ordinarily varies from about 55" to 45" and since the
computation for FQ is very simple at k5”, an effective Eg is selected as
the value corresponding to the value of EG a.t 45”. The total A,E is then
calculated by multiplying the Eg at 45” by the total carcass thickness and
the width of the tread. Thus,
A,E = E~(45") * H * BW ( 10)
The shear modulus of the supporting foundation, G, is the shear
modulus of t.he sidewall portion of the tire. Therefore G is approximated
by its value at a location mid-way between the rim and the shoulder, in
Figure 16 this position is noted by the dimension RS, The cord-half-angle,
cl,, is approximated at this position by the "cosine Law":
cosas = g COSB
Where 0 is the cord half-angle at the crown.
The re.Lative accuracy of these a.pproximate moduli is examined by
comparing their values with those used previously for Tires 1, 2, and J
in Table I. These comparisons and t.he resulting effects on the fore-and-.
aft spring constant, Kf, are shown in Table III.
25
TABLE III
COMPARISON OF RESULTS USING APPROXIMATE MODULI
G-From Table I Tire 1 Tire 2
28440 47164 Tire 3
43876 G-Approx. from Eqs. (8) and (11) ego00 48500 45400
AsE-From Table I 779 747 760 AsE-Approx. from Eq. (7) - a = 45” 780 580 533
Kf-l?rom Table II 2503 2286 2944 Kf-Using Approx. A E and G 2530 2070 2520 Kt-Experimental 2530 2300 2780
Axle --A
i
RI
ROB --L
I ROC
Figure 16. Sidewall location for effective G.
26
As can be seen, the comparisons are relatrvely good and since this
model for calculating Kf is simple, it seems justifiable to use these
simpler expressions for the moduli in approximate calculations of the
spring rate Kf.
To summarize the method for calculating Kf and to illustrate the
relative ease with which it can be done if one uses the approximate moduli
outlined above, an example problem is given below.
PROBLEM STATEMENT
Calculate an approximate fore-and-aft spring rate for the tire described
in Figure 17. This tire roughly corresponds to a standard 9.50 by 14
four-ply tire.
The elastic properties of the individual ply are calculated first
by referring to Eqs. (9):
hH = qLo28)
(.033) = 0.758
h, = $ t (.028)(20) = 0.500
EX = (300)(20)/(.033) = 1.82 x lo5 psi
Ey = 550 [1+2.9( l-o.5oo )] 0.758 x 0.500
= 176opsi
Gw = 705(0.7854)(0.758)(0.500) + 180 (l-0.758 x 0.500)
= 325 psi
The extension modulus Eg is then computed from Eq. (7), using (1: = 45":
27
AESUBC =300 I
ERUB =55Opsi
BW = 5.35 in.
ROC = 14.25 in.
DlAMC = 0.028
Axle
lb1 in.
in.
NCORD =20 ends/in.
GRUB = 18Opsi
P = 37.5O
RI =13.30 in.
a = 14.83 in.
TPLY =0.033 in.
H = 0.132 in.
ROB = 7.30 in.
A = 9.70 in.
Figure 17. Description of hypothetical tire for example problem.
28
E 8
= 4(325) = 1300 psi
Thus A,E can be found from Eq. (10):
A,E = 1300(0.132)(5.35) = 915 lb
The next property to calculate is G, whose mean value is found at a
location approximately half way between the rim and the shoulder. The
radial location RS is determined by locating the point half way between
ROB and RI (see Figure 16), which for this example is:
-___ RI ROB + -- 7.30 +
RS = = 13.30 - = 10.30 in. 2 2
The cord half-angle at this locat,ion is found from Eq. (11):
cos as = 1o-3o cos 37.5” = 0.573 iK3-5.
and sin oS = 0,819. Thus from Eq, (8):
G = ~"82 x 105(o,573)2(0.81g)2 = 4.00 x lo4 psi
Now referring to Eq. (6), the fore -and-aft spring rate is
Kf = 2(AsE)(q)(tanh q?:a)
The value for q is found from Eq. (1) and the definition of KS:
KS = y = z(4.00 X 10’) (O-132) = lo7o lb/in. /in 9* 90
.
9 = J 1070 -m
= l.O@/in.
29
Therefore, the spring rate is
Kf = 2(915)(1.08)(l) = 1980 lb/in,
VI. APPENDIX II
Consider a plane element made up of a series of parallel cords, with
end count n, each carrying a tension load To as shown below.
To sin o * n cos Q.
If the thickness of the lamina is h, then the shear stress 7, which by
equilibrium acts on all edges equally, is
I- = To g sin a cos a (12)
Now consider deformation of the element shown above in the shear direc-
tion, based on the concept that the cord will elongate under tension To.
From the geometry,
x sin cx = 12.8,
where E is cord strain, Jo the original cord length.
x = e.k?o/sin 0 .
31
The number of cords per unit length
on the upper and lower faces is n *
cos a. The tension component per
cord tangential to the upper or
lower faces is To sin CL Hence,
the shear force per unit length
is
But shear strain is defined as
X El? E 7 = a,cos a = .L?, sin 0 cos cz = sin cz cos a (13)
Also, cord tension and strain are related by the cord spring rate
TO = (AE), . E
Hence, shear modulus G is
G=;= To x sin cX cos ~2
sin Q: cos Q; TO
/ (AE)c
- (M:),fsin2 a cos 2cz (14)
But the extension modulus parallel to the cords is (AE), f = Ex, so
that
G = E, sin2 Q: cos2 cz
32
(15)
VII. ACKNOWLEDGMF#TS
The authors wish to thank Mr. B. Bourland, Mr. P.- A. Schultz, and
Mr. B. Bowman for their assistance in obtaining the experimental data pre-
sented in this report.
33
- - -.
VIII. REFEFGXNCES
1. Clark, S. K., "Internal Characteristics of Orthctropic Laminates," Textile Research Journal, Vol. 33, No. 11, Nov. 1963.
2. Clark, S. K., Dodge, R. N., Field, N. L., "Calculation of Elastic Constants of a Single Sheet of Rubber-Coated Fabric," The Univ. of Michigan, ORA Technical Report 02957-14-T, Feb. 1962.
3. Smiley, F. F., and Horne, W. B., "Mechanical Properties of Pneumatic Tires with Special Reference to Modern Aircraft Tires," NACA Technical Note 4110, National Advisory Committee for Aeronautics, Washington, D .i., January, 1958.
4. Clark, S. K., "Simple Approximations for Force-Deflection Characteristics of Aircraft Tires," NASA Contractor Report, NASA CR-439, Washington, D.C., my, 1966.
5. Dodge, R. N., "Prediction of Pneumatic Tire Characteristics from a Cylindrical Shell Model." Paper presented at SAE Mid-Year Meeting, Chicago, ky 1965.
6. Clark, S. K., "An Analog for the Static Loading of a Pneumatic Tire." Paper presented at SAE Mid-Year Meeting, Chicago, May 1965.
7. Tielking, J. T., "Plane Vibration Characteristics of a Pneumatic Tire Model. It Paper presented at SAE Mid-.Year Meeting, Chicagc, May 1965.
34 NASA-Langley, 1967 - 32 CR-900
